The existence of \(R^\pm _0(\lambda )\)  as bounded operators from \(L^2_s\)  to \(L^2_{-s}\)  with \(s>1/2\)  follows from the limiting absorption principle for the resolvent \((-\partial _x^2-z)^{-1}\)  of the free Schrödinger operator \(-\partial _x^2\)  (see e.g. [1]}) and the formula
\(R_0(z)=\frac{1}{2\sqrt{z}}\left[(-\partial _x^2-\sqrt{z})^{-1}-(-\partial _x^2+\sqrt{z})^{-1}\right],\quad z\in {\mathbb {C}}\setminus [0,\infty ),\) 
